Axioms of stereometry

The stereometry axiom system consists of planimetry axioms and the following 3:

1. Whatever the plane, there are points belonging to this plane, and points that do not belong to it.
2. If two different planes have a common point, then they intersect in a straight line passing through this point.
3. If two different straight lines have a common point, then a plane can be drawn through them, and moreover only one.

Stereometry is a branch of geometry that studies figures in space. Its axioms developed as an extension of plane geometry (geometry on a plane), beginning with the works of Euclid in his Elements (c. 300 BC). However:

Euclid did not formulate the axioms of stereometry separately—they were implicit.

In the 19th century, with the development of the axiomatic method (Hilbert, Peano), it became necessary to strictly separate the axioms of space from the axioms of plane geometry.

The modern system of axioms of stereometry is a logical extension of plane geometry, adding a minimal set of statements necessary to describe three-dimensional space.

How to understand each axiom of stereometry

Here is how three axioms of stereometry can be interpreted:

1. Existence of points outside the plane
No matter what the plane, there are points that belong to it and points that do not.

This is the axiom of three-dimensionality: it states that space cannot be reduced to a single plane.

Without it, everything would be two-dimensional—like a sheet of paper.

It guarantees the presence of "depth"—a third dimension.

2. Intersection of Planes
If two distinct planes have a common point, then they intersect along a line passing through that point.

This is the axiom of intersection structure: it describes how planes interact.

Common point → common line. This is important for constructions, proofs, and understanding of geometric configurations.

3. Definiteness of a Plane by Two Intersecting Lines
If two distinct lines have a common point, then a plane—and only one—can be drawn through them.

This is the axiom of uniqueness for a plane passing through two intersecting lines.

It allows one to construct planes from given elements and guarantees uniqueness.

Applications of the Axioms of Stereometry

Stereometry and its axioms underlie many disciplines and practices:

Area Application

• Architecture and Construction Calculating volumes, angles, and structural intersections
• Engineering and Mechanical Engineering Modeling of parts, spatial tolerances
• Chemistry and Crystallography Spatial structure of molecules and lattices
• Computer Graphics and Games 3D modeling, rendering, object intersections
• Mathematics and Education Developing spatial thinking
• Aerospace Technologies Trajectories, orientation of objects in space

How to Meaningfully Study Axioms

Here are some strategies:

• Visualize: draw diagrams, imagine planes and points in space.
• Compare with plane geometry: look for analogies and differences.
• Practice: solve problems, build models, work with CAD systems.
• Think logically: axioms are the foundation from which theorems are derived. Understanding their role is the key to deep thinking.